On a $q$-Skew Amitsur's Theorem

Aristide F. J. -C. Launois

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arXiv · 2605.08100

On a $q$-Skew Amitsur's Theorem

Aristide F. J. -C. Launois

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Abstract

Let $R$ be an algebra over an uncountable field, $\sigma$ a locally torsion automorphism and $\delta$ a locally nilpotent left $\sigma$-derivation such that $q\sigma\delta = \delta\sigma$, where $q$ is a nonzero scalar. We show that the constant part of the Jacobson radical of the Ore extension $R[x;\sigma,\delta]$ is nil. This partially answers a question of Greenfeld, Smoktunowicz and Ziembowski posed in 2019. As a corollary, we employ Shin's 2024 result to prove a q-skew Amitsur's theorem whenever the field is additionally assumed to be of characteristic zero. That is, the Jacobson radical of $R[x;\sigma, \delta]$ is $N[x;\sigma,\delta]$ for some nil ideal $N$ of $R$.

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Aristide F. J. -C. Launois. 2026-04-22. On a $q$-Skew Amitsur's Theorem. https://arxiv.org/abs/2605.08100

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